amuse / bemuse
To entertain versus to bewilder.
To amuse is to entertain or make someone smile. To bemuse is to confuse or leave someone lost in thought. Bemused means puzzled, not mildly entertained. If you mean “quietly amused,” use amused.
Examples
Use The cartoon amused the children.
Use The instructions bemused the new clerk.
Avoid The cartoon left her bemused and grinning. (If you mean she was entertained, the word is amused.)
Some writers use bemused for a faint, wry smile. That sense turns up in print. It still collides with the older meaning.